Problem: Multiply the following complex numbers: $({4+i}) \cdot ({-1+5i})$
Answer: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({4+i}) \cdot ({-1+5i}) = $ $ ({4} \cdot {-1}) + ({4} \cdot {5}i) + ({1}i \cdot {-1}) + ({1}i \cdot {5}i) $ Then simplify the terms: $ (-4) + (20i) + (-1i) + (5 \cdot i^2) $ Imaginary unit multiples can be grouped together. $ -4 + (20 - 1)i + 5i^2 $ After we plug in $i^2 = -1$ , the result becomes $ -4 + (20 - 1)i - 5 $ The result is simplified: $ (-4 - 5) + (19i) = -9+19i $